Chapter 16

Along the curve \(\mathbf{r}(t)=t \mathbf{i}-\mathbf{j}+t^{2} \mathbf{k}, 0 \leq t \leq 1,\) evaluate each of the following integrals. \begin{equation} \begin{array}{l}{\text { a. } \int_{C}(x+y-z) d x \quad \text { b. } \int_{C}(x+y-z) d y} \\ {\text { c. } \int_{C}(x+y-z) d z}\end{array} \end{equation}

Integrate \(f ( x , y , z ) = ( x + y + z ) / \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right)\) over the path \(\mathbf { r } ( t ) = t \mathbf { i } + t \mathbf { j } + t \mathbf { k } , 0 < a \leq t \leq b\)

In Exercises 9- \(20,\) use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D .\) Thick sphere \(\quad \mathbf{F}=(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) / \sqrt{x^{2}+y^{2}+z^{2}}\) \(D :\) The region \(1 \leq x^{2}+y^{2}+z^{2} \leq 4\)

In Exercises \(17-26,\) use a parametrization to express the area of the surface as a double integral. Then evaluate the integral. (There are many correct ways to set up the integrals, so your integrals may not be the same as those in the back of the book. They should have the same values, however.) $$ \begin{array}{l}{\text { Plane inside cylinder The portion of the plane } z=-x \text { inside }} \\ {\text { the cylinder } x^{2}+y^{2}=4}\end{array} $$

Although they are not defined on all of space \(R^{3},\) the fields associated with Exercises \(18-22\) are conservative. Find a potential function for each field and evaluate the integrals as in Example \(6 .\) $$\int_{(0,2,1)}^{(1, \pi / 2,2)} 2 \cos y d x+\left(\frac{1}{y}-2 x \sin y\right) d y+\frac{1}{z} d z$$

In Exercises \(9-20\) , use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D .\) Thick sphere \(\quad \mathbf{F}=\left(5 x^{3}+12 x y^{2}\right) \mathbf{i}+\left(y^{3}+e^{y} \sin z\right) \mathbf{j}+\) \(\left(5 z^{3}+e^{y} \cos z\right) \mathbf{k}\) \(D :\) The solid region between the spheres \(x^{2}+y^{2}+z^{2}=1\) and \(x^{2}+y^{2}+z^{2}=2\)

In Exercises \(19-24,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(F\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\) . \begin{equation} \begin{array}{l}{\mathbf{F}=2 \mathbf{z}+3 x \mathbf{j}+5 y \mathbf{k}} \\ {S : \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+\left(4-r^{2}\right) \mathbf{k}} \\ {0 \leq r \leq 2, \quad 0 \leq \theta \leq 2 \pi}\end{array} \end{equation}

Use Green's Theorem to find the counterclockwise circulation and outward flux for the field \(\mathbf{F}\) and curve \(C .\) \(\mathbf{F}=\left(x+e^{x} \sin y\right) \mathbf{i}+\left(x+e^{x} \cos y\right) \mathbf{j}\) \(C :\) The right-hand loop of the lemniscate \(r^{2}=\cos 2 \theta\)

In Exercises \(19-28,\) use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the specified direction. Parabolic cylinder \(\mathbf{F}=z^{2} \mathbf{i}+x \mathbf{j}-3 z\) outward (normal away from the \(x\) -axis) through the surface cut from the parabolic cylinder \(z=4-y^{2}\) by the planes \(x=0, x=1,\) and \(z=0\)

In Exercises \(19-22,\) find the work done by \(F\) over the curve in the direction of increasing \(t .\) \begin{equation} \begin{array}{l}{\mathbf{F}=x y \mathbf{i}+y \mathbf{j}-y z \mathbf{k}} \\\ {\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1}\end{array} \end{equation}

Evaluate \(\int _ { C } x d s ,\) where \(C\) is a. the straight-line segment \(x = t , y = t / 2 ,\) from \(( 0,0 )\) to \(( 4,2 ) .\) b. the parabolic curve \(x = t , y = t ^ { 2 } ,\) from \(( 0,0 )\) to \(( 2,4 )\)

In Exercises 9-20, use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D\) . Thick cylinder \(\quad \mathbf{F}=\ln \left(x^{2}+y^{2}\right) \mathbf{i}-\left(\frac{2 z}{x} \tan ^{-1} \frac{y}{x}\right) \mathbf{j}+\) \(z \sqrt{x^{2}+y^{2}} \mathbf{k}\) \(D :\) The thick-walled cylinder \(1 \leq x^{2}+y^{2} \leq 2,-1 \leq z \leq 2\)

In Exercises \(19-24,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(F\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\) . \begin{equation} \begin{array}{l}{\mathbf{F}=(y-z) \mathbf{i}+(z-x) \mathbf{j}+(x+z) \mathbf{k}} \\ {S : \quad \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+\left(9-r^{2}\right) \mathbf{k},} \\ {0 \leq r \leq 3, \quad 0 \leq \theta \leq 2 \pi}\end{array} \end{equation}

In Exercises \(17-26,\) use a parametrization to express the area of the surface as a double integral. Then evaluate the integral. (There are many correct ways to set up the integrals, so your integrals may not be the same as those in the back of the book. They should have the same values, however.) $$ \begin{array}{l}{\text { Cone frustum The portion of the cone } z=\sqrt{x^{2}+y^{2}} / 3 \text { between }} \\ {\text { the planes } z=1 \text { and } z=4 / 3}\end{array} $$

Use Green's Theorem to find the counterclockwise circulation and outward flux for the field \(\mathbf{F}\) and curve \(C .\) \(\mathbf{F}=\left(\tan ^{-1} \frac{y}{x}\right) \mathbf{i}+\ln \left(x^{2}+y^{2}\right) \mathbf{j}\) \(C :\) The boundary of the region defined by the polar coordinate inequalities \(1 \leq r \leq 2,0 \leq \theta \leq \pi\)

In Exercises \(19-28,\) use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the specified direction. Parabolic cylinder \(\quad \mathbf{F}=x^{2} \mathbf{j}-x z \mathbf{k}\) outward (normal away from the \(y z\) -plane) through the surface cut from the parabolic cylinder \(y=x^{2},-1 \leq x \leq 1,\) by the planes \(z=0\) and \(z=2\)

In Exercises \(19-22,\) find the work done by \(F\) over the curve in the direction of increasing \(t .\) \begin{equation} \begin{array}{l}{\mathbf{F}=2 \mathrm{yi}+3 x \mathbf{j}+(x+y) \mathbf{k}} \\\ {\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+(t / 6) \mathbf{k}, \quad 0 \leq t \leq 2 \pi}\end{array} \end{equation}

Evaluate \(\int _ { C } \sqrt { x + 2 y } d s ,\) where \(C\) is a. the straight-line segment \(x = t , y = 4 t ,\) from \(( 0,0 )\) to \(( 1,4 )\) . b. \(C _ { 1 } \cup C _ { 2 } ; C _ { 1 }\) is the line segment from \(( 0,0 )\) to \(( 1,0 )\) and \(C _ { 2 }\) is the line segment from \(( 1,0 )\) to \(( 1,2 )\) .

a. Show that the outward flux of the position vector field \(\mathbf{F}=\) \(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) through a smooth closed surface \(S\) is three times the volume of the region enclosed by the surface. b. Let n be the outward unit normal vector field on \(S .\) Show that it is not possible for \(F\) to be orthogonal to \(n\) at every point of \(S .\)

In Exercises \(19-24,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(F\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\) . \begin{equation} \begin{array}{l}{\mathbf{F}=x^{2} y \mathbf{i}+2 y^{3} z \mathbf{j}+3 \mathbf{k}} \\ {S : \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+r \mathbf{k}} \\ {0 \leq r \leq 1, \quad 0 \leq \theta \leq 2 \pi}\end{array} \end{equation}

Find the counterclockwise circulation and outward flux of the field \(\mathbf{F}=x y \mathbf{i}+y^{2} \mathbf{j}\) around and over the boundary of the region enclosed by the curves \(y=x^{2}\) and \(y=x\) in the first quadrant.

In Exercises \(17-26,\) use a parametrization to express the area of the surface as a double integral. Then evaluate the integral. (There are many correct ways to set up the integrals, so your integrals may not be the same as those in the back of the book. They should have the same values, however.) $$ \begin{array}{l}{\text { Circular cylinderband The portion of the cylinder } x^{2}+y^{2}=1} \\ {\text { between the planes } z=1 \text { and } z=4}\end{array} $$

In Exercises \(19-28,\) use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the specified direction. Sphere \(\quad \mathbf{F}=z \mathbf{k}\) across the portion of the sphere \(x^{2}+y^{2}+\) \(z^{2}=a^{2}\) in the first octant in the direction away from the origin

In Exercises \(19-22,\) find the work done by \(F\) over the curve in the direction of increasing \(t .\) \begin{equation} \begin{array}{l}{\mathbf{F}=z \mathbf{i}+x \mathbf{j}+y \mathbf{k}} \\\ {\mathbf{r}(t)=(\sin t) \mathbf{i}+(\cos t) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 2 \pi}\end{array} \end{equation}

Find the line integral of \(f ( x , y ) = y e ^ { x ^ { 2 } }\) along the curve \(\mathbf { r } ( t ) = 4 t \mathbf { i } - 3 t \mathbf { j } , - 1 \leq t \leq 2\)

The base of the closed cubelike surface shown here is the unit square in the \(x y\) -plane. The four sides lie in the planes \(x=0\) , \(x=1, y=0,\) and \(y=1 .\) The top is an arbitrary smooth surface whose identity is unknown. Let \(\mathbf{F}=x \mathbf{i}-2 y \mathbf{j}+(z+3) \mathbf{k}\) and suppose the outward flux of \(\mathbf{F}\) through Side \(A\) is 1 and through Side \(B\) is \(-3 .\) Can you conclude anything about the outward flux through the top? Give reasons for your answer.

In Exercises \(19-24,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(F\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\) . \begin{equation} \begin{array}{l}{\mathbf{F}=(x-y) \mathbf{i}+(y-z) \mathbf{j}+(z-x) \mathbf{k}} \\ {S : \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+(5-r) \mathbf{k}} \\ {0 \leq r \leq 5, \quad 0 \leq \theta \leq 2 \pi}\end{array} \end{equation}

Find the counterclockwise circulation and the outward flux of the field \(\mathbf{F}=(-\sin y) \mathbf{i}+(x \cos y) \mathbf{j}\) around and over the square cut from the first quadrant by the lines \(x=\pi / 2\) and \(y=\pi / 2\) .

In Exercises \(17-26,\) use a parametrization to express the area of the surface as a double integral. Then evaluate the integral. (There are many correct ways to set up the integrals, so your integrals may not be the same as those in the back of the book. They should have the same values, however.) $$ \begin{array}{l}{\text { Circular cylinder band The portion of the cylinder } x^{2}+z^{2}=} \\ {10 \text { between the planes } y=-1 \text { and } y=1}\end{array} $$

In Exercises \(19-28,\) use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the specified direction. Sphere \(\mathbf{F}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) across the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) in the direction away from the origin

In Exercises \(19-22,\) find the work done by \(F\) over the curve in the direction of increasing \(t .\) \begin{equation} \begin{array}{l}{\mathbf{F}=6 z \mathbf{i}+y^{2} \mathbf{j}+12 x \mathbf{k}} \\\ {\mathbf{r}(t)=(\sin t) \mathbf{i}+(\cos t) \mathbf{j}+(t / 6) \mathbf{k}, \quad 0 \leq t \leq 2 \pi}\end{array} \end{equation}

Find the line integral of \(f ( x , y ) = x - y + 3\) along the curve \(\mathbf { r } ( t ) = ( \cos t ) \mathbf { i } + ( \sin t ) \mathbf { j } , 0 \leq t \leq 2 \pi\)

Let \(\mathbf{F}=(y \cos 2 x) \mathbf{i}+\left(y^{2} \sin 2 x\right) \mathbf{j}+\left(x^{2} y+z\right) \mathbf{k} .\) Is there a vector field \(\mathbf{A}\) such that \(\mathbf{F}=\nabla \times \mathbf{A} ?\) Explain your answer.

In Exercises \(19-24,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(F\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\) . \begin{equation} \begin{array}{l}{\mathbf{F}=3 y \mathbf{i}+(5-2 x) \mathbf{j}+\left(z^{2}-2\right) \mathbf{k}} \\ {S : \quad \mathbf{r}(\phi, \theta)=(\sqrt{3} \sin \phi \cos \theta) \mathbf{i}+(\sqrt{3} \sin \phi \sin \theta) \mathbf{j}+} \\ {(\sqrt{3} \cos \phi) \mathbf{k}, \quad 0 \leq \phi \leq \pi / 2, \quad 0 \leq \theta \leq 2 \pi}\end{array} \end{equation}

In Exercises \(19-28,\) use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the specified direction. \(\mathbf{F}\) lane \(\mathbf{F}=2 x y \mathbf{i}+2 y z \mathbf{j}+2 x z \mathbf{k}\) upward across the portion of the plane \(x+y+z=2 a\) that lies above the square \(0 \leq x \leq a\) \(0 \leq y \leq a,\) in the \(x y-\) plane

Evaluate \(\int_{C} x y d x+(x+y) d y\) along the curve \(y=x^{2}\) from \((-1,1)\) to \((2,4)\)

Evaluate \(\int _ { C } \frac { x ^ { 2 } } { y ^ { 4 / 3 } } d s ,\) where \(C\) is the curve \(x = t ^ { 2 } , y = t ^ { 3 } ,\) for \(1 \leq t \leq 2\)

Find the outward flux of the field $$\mathbf{F}=\left(3 x y-\frac{x}{1+y^{2}}\right) \mathbf{i}+\left(e^{x}+\tan ^{-1} y\right) \mathbf{j}$$ across the cardioid \(r=a(1+\cos \theta), a>0.\)

Outward flux of a gradient field Let \(S\) be the surface of the portion of the solid sphere \(x^{2}+y^{2}+z^{2} \leq a^{2}\) that lies in the first octant and let \(f(x, y, z)=\ln \sqrt{x^{2}+y^{2}}+z^{2}\) . Calculate $$\iint_{S} \nabla f \cdot \mathbf{n} d \sigma$$ \((\nabla f \cdot \mathbf{n}\) is the derivative of \(f\) in the direction of outward normal \(\mathbf{n} .)\)

Find the counterclockwise circulation of \(\mathbf{F}=\left(y+e^{x} \ln y\right) \mathbf{i}+\) \(\left(e^{x} / y\right) \mathbf{j}\) around the boundary of the region that is bounded above by the curve \(y=3-x^{2}\) and below by the curve \(y=x^{4}+1.\)

In Exercises \(19-24,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(F\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\) . \begin{equation} \begin{array}{l}{\mathbf{F}=y^{2} \mathbf{i}+z^{2} \mathbf{j}+x \mathbf{k}} \\\ {S : \quad \mathbf{r}(\phi, \theta)=(2 \sin \phi \cos \theta) \mathbf{i}+(2 \sin \phi \sin \theta) \mathbf{j}+(2 \cos \phi) \mathbf{k}} \\\ {0 \leq \phi \leq \pi / 2, \quad 0 \leq \theta \leq 2 \pi}\end{array} \end{equation}

Evaluate $$\int_{C} x^{2} d x+y z d y+\left(y^{2} / 2\right) d z$$ along the line segment \(C\) joining \((0,0,0)\) to \((0,3,4)\).

In Exercises \(17-26,\) use a parametrization to express the area of the surface as a double integral. Then evaluate the integral. (There are many correct ways to set up the integrals, so your integrals may not be the same as those in the back of the book. They should have the same values, however.) $$ \begin{array}{l}{\text { Parabolic band The portion of the paraboloid } z=x^{2}+y^{2} \text { between }} \\ {\text { the planes } z=1 \text { and } z=4}\end{array} $$

In Exercises \(19-28,\) use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the specified direction. Cylinder \(\mathbf{F}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) outward through the portion of the cylinder \(x^{2}+y^{2}=1\) cut by the planes \(z=0\) and \(z=a\)

Evaluate \(\int_{C}(x-y) d x+(x+y) d y\) counterclockwise around the triangle with vertices \((0,0),(1,0),\) and \((0,1) .\)

Find the line integral of \(f ( x , y ) = \sqrt { y } / x\) along the curve \(\mathbf { r } ( t ) = t ^ { 3 } \mathbf { i } + t ^ { 4 } \mathbf { j } , 1 / 2 \leq t \leq 1\)

Let \(\mathbf{F}\) be a field whose components have continuous first partial derivatives throughout a portion of space containing a region \(D\) bounded by a smooth closed surface \(S .\) If \(|\mathbf{F}| \leq 1,\) can any bound be placed on the size of $$\iiint_{D} \nabla \cdot \mathbf{F} d V ?$$ Give reasons for your answer.

Find the work done by \(\mathbf{F}\) in moving a particle once counterclockwise around the given curve. \(\mathbf{F}=2 x y^{3} \mathbf{i}+4 x^{2} y^{2} \mathbf{j}\) C: The boundary of the "triangular" region in the first quadrant enclosed by the \(x\) -axis, the line \(x=1,\) and the curve \(y=x^{3}\)

Let \(C\) be the smooth curve \(\mathbf{r}(t)=(2 \cos t) \mathbf{i}+(2 \sin t) \mathbf{j}+\) \(\left(3-2 \cos ^{3} t\right) \mathbf{k},\) oriented to be traversed counterclockwise around the \(z\) -axis when viewed from above. Let \(S\) be the piecewise smooth cylindrical surface \(x^{2}+y^{2}=4,\) below the curve for \(z \geq 0,\) together with the base disk in the \(x y\) -plane. Note that \(C\) lies on the cylinder \(S\) and above the \(x y\) -plane (see the accompanying figure). Verify Equation \((4)\) in Stokes' Theorem for the vector field \(\mathbf{F}=y \mathbf{i}-x \mathbf{j}+x^{2} \mathbf{k.}\)

Evaluate \(\int_{C} \mathbf{F} \cdot \mathbf{T} d s\) for the vector field \(\mathbf{F}=x^{2} \mathbf{i}-y \mathbf{j}\) along the curve \(x=y^{2}\) from \((4,2)\) to \((1,-1)\) .